Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting | Q82 ⋊ C3⋊D3 |
| derived | SL(2,𝔽3)2 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | A4⋊S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3)2 | C2 |
| Fitting | Q82 | C3⋊D3 |
| LCS | G ▷ SL(2,𝔽3)2 | |
| UCS | C1 ◁ C22 | |
248 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2,3^2,2^6|d^2=i,e^2=i,f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=dei,e^a=ei,f^a=fgh,g^a=gh,f^b=fg,g^b=f,d^c=de,e^c=d,e^d=ei,g^f=gh38 conjugacy classes — view the full character table.
| maximal subgroup of | #72244547, GL(2,𝔽3)2 |
| maximal normal subgroup of | GL(2,𝔽3)2 |
| maximal quotient of | #72244547 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155477);